Hot air balloon mass including increase in mass with balloon volume

[Finn_J]

Homework Statement

I'm currently writing a paper for university where I am producing an outline for a UAV to explore Titan, its design is a hybrid between a hot air balloon and a glider. It stays afloat using the buoyancy of its wing which contains gas from Titans atmosphere heated by an RTG to displace its mass.

Relevant Equations

I am using the equation:
volume = M(mass to be displaced)/|internal gas density - external gas density| = M/dD

Hi guys,

M can then be split up into 2 components: the mass of the payload which is known, and the mass of the wing.

The wing material has a known thickness and density meaning its mass is given by: surface area of wing* mass per m^2 (mass m^2 is used as density and thickness are known so an equation for mass per surface area can be used) :

This means that I can get the equation:
V = Mp/dD + Mw(V)/dD
Where; V is volume of the wing, Mp is the mass of the payload, dD is the density difference of internal and external atmosphere, and Mw(V) is the mass of the wing given by the formula: Mw(V) = surface area of wing* mass per m^2 = Sa*Ds

Mw(V) can then be written as: Ds*V/D, where D is the depth of the wing (the wing will be a rectangular based pyramid and the depth is known).

This is where I think I start to make a mistake:
I then put this equation in integral form to try and end up with a quadratic in V that can be solved to give a value for V. To start I get the equation:
V = Mp/dD + Ds/(D*dD) * intergral(V dV)

Resulting in: V = Mp/dD + Ds*V^2/(D*dD)
Rearranged to give: V^2(Ds/D*dD) - V + Mp/dD

Solving this quadratic gives me complex results which obviously I can not use as a volume value. Any help on this would be much apricated as I have been stuck on this for a while now. I understand that this question is a little bit specific and messy so please don't hesitate to ask me to re-define something.

Many thanks!

 

[kuruman]

I am confused about what you already know (and consider given) and what it is that you wish to know. Can you clarify that a bit more? Obviously you are looking for the volume V of the wing. What are the considerations and constraints that might affect it?

 

[Finn_J]

I am confused about what you already know (and consider given) and what it is that you wish to know. Can you clarify that a bit more? Obviously you are looking for the volume V of the wing. What are the considerations and constraints that might affect it?

Hi sorry I realize I haven't made this very clear, I've done about 7 hours straight of work today and I'm 4 coffees in so my brain isn't working at peak performance.

I know the total mass of the payload, the external temperature and pressure, the internal temperature and pressure, the density of the material to be used in the wing/balloon. I'm trying to find a formula that will give me the total required volume of the balloon in order to displace the mass of the whole probe. However, the mass of the probe is made up of the payload mass (which is known) and the mass of the balloon itself (which depends on volume). The balloon is to be made of polyester coated Mylar so it is very low density, so obviously I could just make an assumption about the mass of the balloon and add that to the mass of the payload, but I'd prefer to have a more precise value.

I attempted to do this by producing an equation that relates volume to the mass of the payload and the mass of the balloon. I then put the mass of the balloon in the form of an integral equation which I solved to get a quadratic that I thought I could solve to get a value for volume. Honestly I don't really know if this is on the right path.

I hope this clarifies things slightly, Thanks!

 

[kuruman]

I'm trying to find a formula that will give me the total required volume of the balloon in order to displace the mass of the whole probe.

If that is all you want to find, then the formula is easy to write down. According to Archimedes, the average density $\rho_{\text{avg.}}$ of whatever it is you want to float must be less than the density of the external fluid or gas. The average density is the total mass, i.e. of payload + wing + gas in wing + etc., divided by the unknown external volume, i.e. whatever displaces external gas. Then $$\rho_{\text{avg.}}=\frac{M_{\text{total}}}{V_{\text{ext.}}} \leq \rho_{\text{ext.}}~\implies~ V_{\text{ext.}}\geq \frac{M_{\text{total}}}{\rho_{\text{ext.}}}. $$In short, the external volume that you are looking for must be greater than the total mass you wish to float divided by the density of the gas (or fluid) surrounding it. You should, of course, allow a margin of safety.

 

[Finn_J]

If that is all you want to find, hen the formula is easy to write down. According to Archimedes, the average density $\rho_{\text{avg.}}$ of whatever it is you want to float must be less than the density of the external fluid or gas. The average density is the total mass, i.e. of payload + wing + gas in wing + etc., divided by the unknown external volume, i.e. whatever displaces external gas. Then $$\rho_{\text{avg.}}=\frac{M_{\text{total}}}{V_{\text{ext.}}} \leq \rho_{\text{ext.}}~\implies~ V_{\text{ext.}}\geq \frac{M_{\text{total}}}{\rho_{\text{ext.}}}. $$In short, the external volume that you are looking for must be greater than the total mass you wish to float divided by the density of the gas (or fluid) surrounding it. You should, of course, allow a margin of safety.

Hi thanks for the response,

I'm aware of Archimedes principle and that is in fact where I started for this equation. The issue comes in because the mass of the balloon is given by its volume so using Archimedes principle in its standard form does not work.

I'm trying to find an equation that relates volume to the mass that needs to be displaced for the payload + the mass that needs to be displaced for the mass of a balloon with the given volume.

Something like:

$V \geq \frac{M_{text{payload}{\rho_{\text{ext.}}} + \frac{M_{text{ballon}(V)}{\rho_{\text{ext.}}}$

I understand this is a very specific issue and I'm not making it very clear. If I can't find a resolution soon I'll just make an assumption about the mass of the balloon as being made from polyester coated Mylar its mass will be a few kg.

Thanks!

 

[kuruman]

$\dots~$so using Archimedes principle in its standard form does not work.

What is this "standard" form that does not work and why? The form I know is that the buoyant force is equal to the weight of the displaced fluid. It follows that if the displaced fluid weighs more than the object doing the displacing, the buoyant force ill lift th objet against gravity. If shipping companies can float ships on the ocean and move them from A to B without worrying about the details of what the payload or the hull material are, why can you not do the same thing?

Perhaps I did not understand what you are trying to accomplish here. Are there constraints that you have not mentioned? Furthermore, the equation that you posted in #3 did not render in LaTeX. I can guess what the first fraction is but not the second so I am still in the dark.

 

[kuruman]

After additional thought, I think I understand what you are trying to say. Let's say you want to float a rectangular box of outer dimensions $a\times b \times c$ and skin thickness $t$ with payload $M$. Let
$\rho_0$ = the density of the fluid in which the box will float
$\rho_{\text{skin}}$ = the density of the material that makes up the skin

Then the box will be neutrally buoyant if the mass of the displaced fluid is equal to the payload mass and the mass of the skin, $$\rho_0 abc=M+\rho_{\text{skin}}V_{\text{skin}}=M+\rho_{\text{skin}}(2abt+2bct+2cat).$$At this point, additional input is needed about the relative dimensions of the box depending on what you want it to look like. Let's say that if $c$ is the smallest side, you want $b=4c$ and $a=10c$ so that it looks like a plank. Then the expression above becomes $$\rho_0 40c^3 = M+2t\rho_{\text{skin}}(ab+bc+ca)=M+2tc^2\rho_{\text{skin}}(40+4+10)=M+108t\rho_{\text{skin}}c^2.$$It seems then that, for this particular design, you need to solve the third degree equation,$$40\rho_0 c^3-108t\rho_{\text{skin}}c^2-M=0.$$ Is your problem not knowing how to solve an equation like this? Note that you will always end up with an equation of the form $$A x^3+B x^2+C=0$$ regardless of the shape of the floating object. That is obvious in the derivation. Probably the best way to solve the cubic equation is by plotting it and looking for zeroes.

 

[Finn_J]

After additional thought, I think I understand what you are trying to say. Let's say you want to float a rectangular box of outer dimensions $a\times b \times c$ and skin thickness $t$ with payload $M$. Let
$\rho_0$ = the density of the fluid in which the box will float
$\rho_{\text{skin}}$ = the density of the material that makes up the skin

Then the box will be neutrally buoyant if the mass of the displaced fluid is equal to the payload mass and the mass of the skin, $$\rho_0 abc=M+\rho_{\text{skin}}V_{\text{skin}}=M+\rho_{\text{skin}}(2abt+2bct+2cat).$$At this point, additional input is needed about the relative dimensions of the box depending on what you want it to look like. Let's say that if $c$ is the smallest side, you want $b=4c$ and $a=10c$ so that it looks like a plank. Then the expression above becomes $$\rho_0 40c^3 = M+2t\rho_{\text{skin}}(ab+bc+ca)=M+2tc^2\rho_{\text{skin}}(40+4+10)=M+108t\rho_{\text{skin}}c^2.$$It seems then that, for this particular design, you need to solve the third degree equation,$$40\rho_0 c^3-108t\rho_{\text{skin}}c^2-M=0.$$ Is your problem not knowing how to solve an equation like this? Note that you will always end up with an equation of the form $$A x^3+B x^2+C=0$$ regardless of the shape of the floating object. That is obvious in the derivation. Probably the best way to solve the cubic equation is by plotting it and looking for zeroes.

Hi, thank you! this is what I am looking for, I see now where I was going wrong. Thank you for the solution it is much simpler that what I was trying to do. I'll start forming by 3rd order equation right away, in order to solve it could I not differentiate it twice and find V when the equation is equal to zero in order to find the stationary points?

Thanks!

 

[kuruman]

$\dots~$ in order to solve it could I not differentiate it twice and find V when the equation is equal to zero in order to find the stationary points?

What stationary points? If you do what you say, you end up with $6Ax+2B=0$. Note that the constant term which is related to the payload mass has disappeared and is no longer an input to the desired solution. Not convinced? Consider the equation, $x^3-7x^2+36=0$. Convince yourself that it has 3 real roots x = {6, 3, -2}. Then try your method and see if you can recover them.

If I were you, I would do some web research on how to solve a cubic equation.